How to find a codimension-one heteroclinic
cycle between two periodic orbits

The concept of Sustainable Development has given rise to multiple interpretations. In this article, it is proposed that Sustainable Development should be interpreted as the capacity of territory, community, or landscape to conserve the notion of well-being that its population has agreed upon. To see the implications of this interpretation, a Brander and Taylor model, to evaluate the implications that extractivist policies have over an isolated community and cooperating communities, is proposed. For an isolated community and through a bifurcation analysis in which the Hopf bifurcation and the heteroclinic cycle bifurcation are detected, 4 prospective scenarios are found, but only one is sustainable under different extraction policies. In the case of cooperation, the exchange between communities is considered by coupling two models such as the one defined for the isolated community, with the condition that their transfers of renewable resources involve conservation policies. Since human decisions do not occur in a continuum, but rather through jumps, the mathematical model of cooperation used is a Filippov System, in which the dynamics could involve two switching manifolds of codimension one and one switching manifold of codimension two. The exchange in the cooperation model, for specific parameter arrangements, exhibits n -periodic orbits and chaos. It is notable that, in the cases in which the system shows sliding, it could be interpreted as a recovery delay related to the time needed by the deficit community to recover, until its dependence on the other community stops. It is concluded (1) that a sustainability analysis depends on the way well-being is defined because every definition of well-being is not necessarily sustainable, (2) that sustainability can be visualized as invariant sets in the nonzero region of the space of states (equilibrium points, n -periodic orbits, and strange attractors), and (3) that exchange is key to the prevalence of the human being in time. The results question us on whether Sustainable Development is only to keep us alive or if it also implies doing it with dignity.

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Links and Bifurcations in Nonsingular Morse–Smale Systems

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127497001345 ◽

1997 ◽

Vol 07 (08) ◽

pp. 1717-1736 ◽

Cited By ~ 1

Author(s):

B. Campos ◽

J. Martínez Alfaro ◽

P. Vindel

Keyword(s):

Periodic Orbits ◽

Codimension One ◽

Nms Systems

Wada's theorem classifies the set of periodic orbits in NMS systems on S3 as links, that can be written in terms of six operations. This characterization allows us to study the topological restrictions that links require to suffer a given codimension one bifurcation. Moreover, these results are reproduced in the case of NMS systems with rotational symmetries, introducing new geometrical tools.

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Chaos in sociobiology

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270001426x ◽

1995 ◽

Vol 51 (3) ◽

pp. 439-451 ◽

Cited By ~ 2

Author(s):

J.R. Christie ◽

K. Gopalsamy ◽

Jibin Li

Keyword(s):

Periodic Orbits ◽

Evolutionary Biology ◽

Mating Behaviour ◽

Continuous Family ◽

Heteroclinic Cycle ◽

Planar System ◽

Perturbed System ◽

Battle Of The Sexes ◽

Game Theoretic ◽

Time Periodic

It is shown that the dynamical game theoretic mating behaviour of males and females can be modelled by a planar system of autonomous ordinary differential equations. This system occurs in modelling “the battle of the sexes” in evolutionary biology. The existence of a heteroclinic cycle and a continuous family of periodic orbits of the system is established; then the dynamical characteristics of a time-periodic perturbation of the system are investigated. By using the well-known Melnikov's method, a sufficient condition is obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Finally, subharmonic Melnikov theory is used to obtain a criterion for the existence of subharmonic periodic orbits of the perturbed system.

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MULTIPARAMETRIC BIFURCATIONS IN AN ENZYME-CATALYZED REACTION MODEL

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127405012521 ◽

2005 ◽

Vol 15 (03) ◽

pp. 905-947 ◽

Cited By ~ 13

Author(s):

E. FREIRE ◽

L. PIZARRO ◽

A. J. RODRÍGUEZ-LUIS ◽

F. FERNÁNDEZ-SÁNCHEZ

Keyword(s):

Periodic Orbits ◽

Dynamical Behavior ◽

Homoclinic Orbits ◽

Reaction Model ◽

Numerical Continuation ◽

Global Bifurcations ◽

Codimension One ◽

Local Bifurcations ◽

Homoclinic Connections ◽

Local And Global Bifurcations

An exhaustive analysis of local and global bifurcations in an enzyme-catalyzed reaction model is carried out. The model, given by a planar five-parameter system of autonomous ordinary differential equations, presents a great richness of bifurcations. This enzyme-catalyzed model has been considered previously by several authors, but they only detected a minimal part of the dynamical and bifurcation behavior exhibited by the system. First, we study local bifurcations of equilibria up to codimension-three (saddle-node, cusps, nondegenerate and degenerate Hopf bifurcations, and nondegenerate and degenerate Bogdanov–Takens bifurcations) by using analytical and numerical techniques. The numerical continuation of curves of global bifurcations allows to improve the results provided by the study of local bifurcations of equilibria and to detect new homoclinic connections of codimension-three. Our analysis shows that such a system exhibits up to sixteen different kinds of homoclinic orbits and thirty different configurations of equilibria and periodic orbits. The coexistence of up to five periodic orbits is also pointed out. Several bifurcation sets are sketched in order to show the dynamical behavior the system exhibits. The different codimension-one and -two bifurcations are organized around five codimension-three degeneracies.

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Dynamics Near the Heterodimensional Cycles with Nonhyperbolic Equilibrium

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418500037 ◽

2018 ◽

Vol 28 (01) ◽

pp. 1850003

Author(s):

Liu Xingbo

Keyword(s):

Periodic Orbits ◽

Vector Fields ◽

Three Dimensional ◽

Transcritical Bifurcation ◽

Heteroclinic Cycle ◽

Dimensional Vector ◽

Parametric Curves ◽

Return Map ◽

Poincaré Return Map

In this paper, bifurcations of heterodimensional cycle with one nonhyperbolic equilibrium and one saddle-focus in three-dimensional vector fields are investigated. We study the interaction of a transcritical bifurcation with a codimension-0/codimension-2 heteroclinic cycle. Based on the construction of a Poincaré return map, we obtain the expressions of parametric curves of homoclinic and heteroclinic connections around the heterodimensional cycle as well as periodic orbits. Furthermore, the configurations of the parametric curves corresponding to different bifurcations are illustrated.

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STOCHASTIC AVERAGING NEAR LONG HETEROCLINIC ORBITS

Stochastics and Dynamics ◽

10.1142/s0219493707001974 ◽

2007 ◽

Vol 07 (02) ◽

pp. 187-228 ◽

Cited By ~ 3

Author(s):

RICHARD B. SOWERS

Keyword(s):

Boundary Layer ◽

Periodic Orbits ◽

Stochastic Averaging ◽

Heteroclinic Orbits ◽

Heteroclinic Cycle ◽

Two Dimensional ◽

Heteroclinic Cycles

We refine some of the bounds of [10]. There, we considered the effect of diffusive perturbations on a two-dimensional ODE with a heteroclinic cycle. We constructed corrector functions for asymptotically "glueing" together behavior of periodic orbits in the boundary layer near the heteroclinic cycle. Here, we adapt the analysis of [10] to allow for "long" heteroclinic cycles.

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